Netlike partial cubes III. The median cycle property

A graph GG has the Median Cycle Property   (MCP) if every triple (u0,u1,u2)(u0,u1,u2) of vertices of GG admits a unique median or a unique median cycle, that is a gated cycle CC of GG such that for all i,j,k∈{0,1,2}i,j,k∈{0,1,2}, if xixi is the gate of uiui in CC, then: {xi,xj}⊆IG(ui,uj){xi,xj}⊆IG(ui,uj) if i≠ji≠j, and dG(xi,xj)